A characterization of 3D steady Euler flows using commuting zero-flux homologies

2019/04/01 by Peralta-Salas, Daniel, Rechtman, Ana, de Lizaur, Francisco Torres
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.00960

Abstract

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.

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• Steady 3d Euler flows via a topology-preserving convex integration scheme
• Steady Compressible 3D Euler Flows in Toroidal Volumes without Continuous Euclidean Isometries
• Isolated steady solutions of the 3D Euler equations
• KAM theory and the 3D Euler equation